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Local uniqueness of an x₀-minimum-norm solution under TCC

Statement

Let $X, Y$ be real Hilbert spaces, let $U \subseteq X$ be open, and let $F : U \to Y$ be Fréchet differentiable. Let $x_{0} \in X$ , $y \in Y$ , and let $x^{†} \in U$ be an $x_{0}$ -minimum-norm solution of $F (x) = y$ on $U$ . Suppose that for some $δ > 0$ the closed ball ${\overline{B}}_{δ} (x^{†})$ lies in $U$ , the local tangential cone condition holds at $x^{†}$ on radius $δ$ with a constant $0 \leq η < 1$ , and

$x^{†} - x_{0} \in \ker (F^{'} (x^{†}))^{⊥} .$

Then every $x \in {\overline{B}}_{δ} (x^{†})$ with $F (x) = y$ and $x \neq x^{†}$ satisfies

$∥ x - x_{0} ∥_{X} > ∥ x^{†} - x_{0} ∥_{X} .$

Consequently, $x^{†}$ is the unique $x_{0}$ -minimum-norm solution among the exact solutions in ${\overline{B}}_{δ} (x^{†})$ .

Statement references

Nonlinear forward maps and exact solutions

The $x_{0}$ -minimum-norm solution condition

Local tangential cone condition

Derivative/data-residual comparison

Fréchet derivative

Kernel of a linear operator

Orthogonal complement

Christian Clason, Regularization of Inverse Problems, Theorem 9.9

Barbara Kaltenbacher, Frank Schöpfer, and Thomas Schuster, Iterative methods for nonlinear ill-posed problems in Banach spaces, Proposition 1

Facts & Assumptions

Given: Real Hilbert spaces $X, Y$ , an open set $U \subseteq X$ , a Fréchet differentiable map $F : U \to Y$ , $x_{0} \in X$ , $y \in Y$ , and an $x_{0}$ -minimum-norm solution $x^{†} \in U$ of $F (x) = y$ . For some $δ > 0$ , ${\overline{B}}_{δ} (x^{†}) \subseteq U$ and the local tangential cone condition holds there with $0 \leq η < 1$ . Also assume $x^{†} - x_{0} \in \ker (F^{'} (x^{†}))^{⊥}$ .

[F1]

Under the local tangential cone condition, $(1 - η) ∥ F (x + h) - F (x) ∥_{Y} \leq ∥ F^{'} (x) h ∥_{Y} \leq (1 + η) ∥ F (x + h) - F (x) ∥_{Y}$ for all admissible $h$ (Tangential cone control compares derivative and data residuals).

[F2]

For a linear map $T : V \to W$ , its kernel is $\ker T : = {v \in V : T (v) = 0_{W}}$ (Kernel and image of a linear map).

[F3]

For a subset $S$ of an inner-product space, $S^{⊥} : = {v : ⟨ v, s ⟩ = 0 for every s \in S}$ (Orthogonality and the orthogonal complement).

[F4]

If $u, v$ are orthogonal in an inner-product space, then $∥ u + v ∥^{2} = ∥ u ∥^{2} + ∥ v ∥^{2}$ (Pythagoras and finite orthogonal sums).

[F5]

The induced Hilbert norm is nonnegative and vanishes only at the zero vector (The induced length is a norm).

[A1]

$x^{†}$ is an exact solution and globally minimizes distance to $x_{0}$ among solutions on $U$ (x₀-minimum-norm solutions of nonlinear equations).

[A2]

The TCC holds at $x^{†}$ on the stated closed ball, with $0 \leq η < 1$ (Local tangential cone condition).

[A3]

$x^{†} - x_{0}$ is orthogonal to every vector in $\ker (F^{'} (x^{†}))$ (Orthogonality and the orthogonal complement).

Proof

technique · direct

1.1givenA2F1F2choosealgebra

Choose any $x \in {\overline{B}}_{δ} (x^{†})$ with $F (x) = y$ and $x \neq x^{†}$ , and put $h : = x - x^{†}$ . Applying [F1] at $x^{†}$ gives $∥ F^{'} (x^{†}) h ∥_{Y} \leq (1 + η) ∥ F (x) - F (x^{†}) ∥_{Y} = 0$ . Hence $F^{'} (x^{†}) h = 0$ , so [F2] gives $h \in \ker (F^{'} (x^{†}))$ ; also $h \neq 0$ .

2.1step 1.1A1A3F3F4F5algebra∎

By [A3] and [F3], $x^{†} - x_{0}$ is orthogonal to $h$ . Applying [F4] to $x^{†} - x_{0}$ and $h$ yields $∥ x - x_{0} ∥_{X}^{2} = ∥ x^{†} - x_{0} ∥_{X}^{2} + ∥ h ∥_{X}^{2} > ∥ x^{†} - x_{0} ∥_{X}^{2}$ , since [F5] and $h \neq 0$ imply $∥ h ∥_{X} > 0$ . The norms are nonnegative, so the strict inequality of squares gives the strict norm inequality. With [A1], this proves uniqueness of the $x_{0}$ -minimum-norm solution among the local exact solutions.

Remarks

The conclusion is local minimum-norm uniqueness, not injectivity of $F$ : other exact solutions may lie in the ball, but each has strictly larger distance to $x_{0}$ .

Depends on

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Sources

Christian Clason, Regularization of Inverse Problems, Theorem 9.9 (standard reference, not scraped)
Barbara Kaltenbacher, Frank Schöpfer, and Thomas Schuster, Iterative methods for nonlinear ill-posed problems in Banach spaces, Proposition 1 (standard reference, not scraped)